Phonon-induced superconducting state on a square lattice

The tendency to induce the superconducting state on a square lattice (the Eliashberg formalism) is characterised in the work. We took into account the pairing mechanism based on a linear electron-phonon interaction. We proved that the superconducting phase cannot exist if the value of the coupling constant in the diagonal channel of the self-energy $\lambda_{D}$ is equal to the value of the coupling constant in the non-diagonal channel $\lambda_{ND}$. The linear electron--phonon interaction induces, however, the superconducting state for values of the unbalance parameter ($\gamma=\lambda_{D}/\lambda_{ND}$) less than $\gamma_{C}=0.42$. The reason for the decrease in the critical temperature value with the growth of the $\gamma$ parameter value is the anomalous increase in the electron effective mass. The results were achieved by the fully self-consistent numerical analysis.